unknown systematic errors and the method of least squares michael grabe
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Unknown systematic errors and the method of least squares Michael Grabe. alternative error model: true values and biases. Quantity to be measured true value. First Principle. Does metrology exist without a net of true values?. Not likely!. Impact of true values and biases - PowerPoint PPT PresentationTRANSCRIPT
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Unknown systematic errors and the method of least squares
Michael Grabe
alternative error model: true values and biases
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Quantity to be measured true value
Does metrology exist without a net of true values?
First Principle
Not likely!
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Impact of true values and biases in least squares
Gauß-Markoff theorem Assessment of uncertainties
Traceability
Key Comparisons
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xAβ
Least squares adjustment
mrmm
r
r
aaa
aaaaaa
...............
...
...
21
22221
11211
A
r
...2
1
β
mx
xx
...2
1
xTraceability
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xAβ xAβ0
m
1iix
m1β
n
1lili xn
1x
Mean of means
averaging is permitted if and only if the respective true values are identical
m
2
1
x...xx
β
1...11
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mass
mg1kg0.25
mg1kg0.75
Adjustment ad hoc ?
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m
2
1
x...xx
empirical variance-covariance matrix
A different approach
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m
1iii xwβ
Mean of means
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xAβ xAβ
m
2
1
x...xx
x
Let the input data be arithmetic means
xAAAβ T1T
00 xAAAβ T1T
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Gauß-Markoff Theorem
The uncertainties are minimal...
...if the system has been weighted appropriately
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biases abolish the theorem ...
according to the GUM we should have
rmQE min rmQmin
but we encounter
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no more test of consistency
how to weight the system to minimize uncertainties?
Consequences ...
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more ... and of utmost importance:
reduce measurement uncertainties
weightings
shift estimators and
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a picture
reduction
before after
shift
true value
kβ
kβ
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Traceability:
vary the weights by trial and error ...
Assessment of uncertainties
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Key ComparisonNational Standards
1β 2β mβ...
true value
true valuetrue value
...
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Round RobinCalibration of a Travelling Standard T
...(1)T (2)T (m)T
(1)β (2)β (m)β...
T
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Key comparisons do more ...
m1,...,i;βTd (i)i
and the differences
Consider the grand mean
(i)m
1iiTwβ
KCRV
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m
1jjs,jis,iis,
m
1j
Tijj
2i
Pd
fwf2wf
wswsw2sn
1ntu
i
βuTuu 2(i)2d i
„consistent“ with
and look forid
(i) uβT where
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Problem:
In some cases the GUM may localize the true value of the travelling standard, in others not ...
whenShould we test (i)T against β
(i)T constributes to β ?
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Differences between KCRV and individual calibrations
1du
2du
mdu
...(2)T
β(m)T
(1)T
true valueKCRV
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Individual Calibrations
a horizontal line should intersect each of the uncertainties
...(1)T
(2)T
(m)Ttrue value
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β KCRV
(1)T
(2)T
...(m)T
true value
KCRV and individual calibrations